Written by Edwin Tunggawan
on July 23, 2020

Hazard Function in Survival Analysis

I’m using this video as my introductory lesson to the hazard function. The hazard function can be defined as follows.

\[h(t) = \frac{f(t)}{S(t)}\]

Given that \(S(t)\) is the survival function (or the reliability function) of the statistical distribution and \(f(t)\) is the probability density function, here’s how to find \(S(t)\).

\[F(t) = \int f(t)\ dt \\ S(t) = 1 - F(t)\]

The hazard function is used to calculate the hazard rate, which is defined as the rate of failure at a certain point in time.

Hazard Rate

The hazard rate can also be called as the failure rate if the context is engineering, the force of mortality if the context is actuaries, or the age-specific death rate in life sciences. This monograph by Professor Horst Rinne from the University of Giessen contains the mathematical construct of the hazard rate function in the first chapter starting on page 9.

For the explanation in this section, I’ll be mainly taking this lecture notes by Associate Professor Rachel MacKay Altman from the Simon Fraser University due to the simplicity.

Given that the probability density function of the distribution is \(f(t)\) and \(X\) is the survival time, the cumulative distribution function can be defined as follows for the probability of failure by the time \(x\).

\[F(x) = P(X \leq x) = \int_{0}^{x} f(t)\ dt\]

As \(F(x)\) is the probability of failure by the time \(x\), we can define the probability of survival by the time \(x\) as follows.

\[S(x) = 1 - F(x)\]

Hence, \(S(x)\) is the survival function or the reliability function of the distribution.

Since the hazard rate is defined as the rate of failure \(f(t)\) at a certain point in time \(x\) for the total surviving population \(S(t)\) at time \(x\), we can calculate the hazard rate for the point in time \(x\) as follows.

\[h(x) = \frac{f(x)}{S(x)}\]

As the time \(X\) goes on, the number of the surviving population \(S(x)\) will become smaller as more individuals from the population have died off.

Cumulative Hazard Function

Cumulative hazard function is defined as follows.

\[H(t) = \int h(t)\ dt\]

Given the time period \(X\), our cumulative hazard function for the time \(x\) within the period \(X\) would be as follows.

\[H(x) = \int_{0}^{x} h(t)\ dt\]

The cumulative hazard function isn’t a probability function with the range of output from 0 to 1, as we can see here the value can surpass 1. But it can be used as a measure of risk. The greater the value of \(H(x)\) for the point in time \(x\) translates to the greater risk of failure by the time \(x\). The problem is that it’s harder to interpret as mentioned in this page of the engineering statistics handbook by NIST.

The book An Introduction to Survival Analysis Using Stata, Second Edition, which is available on Google Books, explains how to interpret cumulative hazard on page 13-15. From the book’s explanation, it seems that the cumulative hazard is a measure of unit of risk of failure within a period of time.